A 3 can be rolled only by tossing a 1-2, but the loaded die can be either the 1 or the 2; similarly, an 11 can be rolled only by tossing a 5-6, but the loaded die can be either the 5 or the 6. We know only the probability of the loaded die coming up 2 and that of it coming up 5; we do not have this information for 1 or 6. We explore two cases and compare the probabilities of rolling a 3 and rolling an 11.

Case 1: Suppose the loaded die will come up 1 with probability 0.1 and 6 with probability 0.3. The probabilities of rolling a 3 or a 11 are, respectively:

and

making 11 the more likely outcome.

Case 2: Suppose the loaded die will come up 1 with probability 0.3 and 6 with probability 0.1. The probabilities of rolling a 3 or a 11 are, respectively:

and 

making 3 the more likely outcome.

Therefore, the two statements together provide insufficient information to answer the question.

Let  be the balls that are completely red, completely yellow, and bicolored, respectively; let  be the total number of balls. Then the probabilities of drawing a ball with some red and drawing a ball with some yellow are, respectively,

 and 

For the former probability to be greater than the latter, we can use some algebra to reveal:

That is, the former probability is greater than the latter if and only if there are more completely red balls than yellow balls. Similarly, the former probability is less than the latter if and only if there are fewer completely red balls than yellow balls. 

Therefore, Statement 1 is irrelevant; Statement 2, which compares the number of completely red balls and completely yellow balls, gives us our necessary and sufficient information.

The difference of the two dice will be 1 in case any of the following outcomes occur:

This is ten outcomes out of a possible 36, so this yields a probability of 

.

Remember, you do not have to SOLVE the data sufficiency problems.  In other words, you do not have to calculate the exact probability of getting a kirsch-flavored (cherry liquor) treat.  You just need to determine how many of the statements are NECESSARY to derive the answer.

The first statement by itself is not sufficient, because knowing that half of the treats are peanut butter flavored does not tell you the proportion that are kirsch flavored.

The second statement is not sufficient, because comparing the proportion of kirsch treats to peanut-butter treats won't help you find the probability of getting a kirsch treat unless you know the ratio of peanut-butter treats in the case.

But, if we put the two pieces of information together, we can solve the problem.  Statement 1 tells us the proportion of peanut-butter treats, and statement 2 tells us how many kirsch treats there are compared to peanut-butter treats.

Therefore, the correct answer is C.

Let P(A) be the probability of Anna picking an apple and P(O) be the probability of Anna picking an orange. We need to evaluate each statement individually first:

(1) gives us the total number of fruits, but no indication as to the number of each fruit, so we cannot determine probabilities form (1) alone

(2) gives us the proportion of oranges. Let N be the total number of fruits, A the number of apples and O the number of oranges. We have:

 We also know that .

We can rewrite this last equation as follow:

 or 

We know that P(A) = A / N and from statement (2) we get

 which we can simplify to find P(A)

Therefore the right answer is B.

Using Statement (1), we can write the following equation:

Let x be the number of left handed people in the sample. 100-x is then the number of right handed people in the sample.

Therefore, the probability that a randomly selected person in the sample is left handed is:

Therefore Statement (1) is sufficient to answer the question

Statement (2) indicates that all women in the sample are right handed. This statement is not useful as it does not give the total number of right handed people in the sample.

Therefore, Statement (2) is not sufficient by itself to answer the question.

Only Statement (1) ALONE is sufficient.

(1) One third of the managers in the company are women.

Therefore the probability that the manager selected at random is a woman is .

Statement (1) alone is sufficient.

(2) Half of the employees in the company are women.

Even though half of the company's employees are women, this statement doesn't give the proportion of women who hold a manager job. Therefore, Statement (2) is not sufficient.

(1) 150 Employees have a MBA degree.

Since we do not know the number of employees at the company, we cannot determine the probability of employees who have an MBA degree. Statement (1) is not sufficient.

(2) 70 percent of employees do not have a MBA degree.

We can then determine the percentage of employees with an MBA degree as 30 percent. Therefore the probability of employees who have an MBA degree is 0.3.

Statement (2) ALONE is sufficient.

(1) Of all 1200 policyholders, 1050 did not have an accident.

From statement (1), we can determine that 150 policyholders had an accident. We can then calculate the probability as .

Statement (1) alone is sufficient.

(2) Of the policyholders who had an accident, 90% received a claim payment.

Statement (2) is irrelevant since we are not interested in the probability of policyholders who received a claim payment.

Statement (2) is not sufficient.

(1) 15 out of 150 light bulbs are defective.

Using statement (1), the probability of defective light bulbs is 

Statement (1) is sufficient.

(2) After checking half of the inventory, the store owner finds 10 defective light bulbs

We do not know the number of defective light bulbs in all the inventory and we do not know the number of light bulbs in inventory.

Statement (2) is not sufficient.